Exact solutions of a linear model of quantum solids

The ground state energyE of a linear quantum crystal can be calculated exactly by the transfer integral method, if the total wave function is assumed to be a product of single-particle functions ? and Jastrow factorsf between nearest neighbors only. The variation ofE with respect tof leads to a Schrödinger equation containing an effective potential, which consists of the bare interaction between two particles and a part due to the influence of all other particles. This equation is coupled to the transfer integral equation. For any given ? the coupled system of equations can be solved in a self-consistent way. The one- and two-particle distribution functions and the effective potential can also be computed exactly. In a similar way for any givenf the optimal ? can be determined.     

Meshless approach to shape optimization of linear thermoelastic solids

This paper presents a formulation for shape optimization in thermoelasticity using a meshless method, namely the element‐free Galerkin method. Two examples are treated in detail and comparisons with previously published finite element analysis results demonstrate the excellent opportunities the EFG offers for solving these types of problems. Smoother stresses, no remeshing, and better accuracy than finite element solutions, permit answers to shape optimization problems in thermoelasticity that are practically unattainable with the classical FEM without remeshing. For the thermal fin example, the EFG finds finger shapes that are missed by the FEM analysis, and the objective value is greatly improved compared to the FEM solution. A study of the influence of the number of design parameters is performed and it is observed that the EFG can give better results with a smaller number of design parameters than is possible with traditional methods. Copyright © 2001 John Wiley & Sons, Ltd.     

非线性奇异振子的解析逼近解

利用修正的牛顿-谐波平衡法建立了非线性奇异振子的解析逼近周期和周期解．通过改写控制方程和选取简单、合适的校正项对牛顿-谐波平衡法进行了修正.构造的两个解析逼近周期和周期解不仅在振幅和参数全部取值范围内有效且能快速地收敛到精确解；两个逼近周期与精确周期的百分比误差分别低于0.92%和0.09%，后者比已有结果精度高。     

Explicit solutions of a moving edge dislocation in multiferroic solids

Dislocation behavior is a critical feature in describing the reliable properties of structures and materials. As a new kind of smart material, multiferroic solids have special applications on the coupling systems of mechanical, electric, and magnetic fields. Then, the moving dislocation in multiferroic solids has special interests for strength characteristics. In the present work, the moving edge dislocation is studied by the eigenstrain theory. The explicit solutions are derived and can be reduced to the piezoelectric and even pure elastic materials as the special cases.     

Discretized peridynamics for linear elastic solids

Peridynamics is a theory of continuum mechanics employing a nonlocal model that can simulate fractures and discontinuities (Askari et?al. J Phys 125:012–078, 2008; Silling J Mech Phys Solids 48(1):175–209, 2000). It reformulates continuum mechanics in forms of integral equations rather than partial differential equations to calculate the force on a material point. A connection between bond forces and the stress in the classical (local) theory is established for the calculation of peridynamic stress, which is calculated by summing up bond forces passing through or ending at the cross section of a node. The peridynamic stress and the constitutive law in elasticity are used for the derivation of one- and three-dimensional numerical micromoduli. For three-dimensional discretized peridynamics, the numerical micromodulus is larger than the analytical micromodulus, and converges to the analytical value as the horizon to grid spacing ratio increases. A comparison of material responses in a three-dimensional discretized peridynamic model using numerical and analytical micromoduli, respectively, is performed for different horizons. As the horizon increases, the boundary effect is more conspicuous, and the errors increase in the back-calculated Young’s modulus and strains. For the simulation of materials of Poisson’s ratios other than 1/4, a pairwise compensation scheme for discretized peridynamics is proposed. Compared with classical (local) elasticity solutions, the computational results by applying the proposed scheme show good agreement in the strain, the resultant Young’s modulus and Poisson’s ratio.     

Analytical solutions to eddy-current testing problems for a layeredmedium with varying properties

A one-parameter family of analytical solutions is found for the case where a single-turn coil is located above a two-layer conducting medium. The conductivity σ and the relative magnetic permeability μ of the upper conducting layer are constants, while for the lower conducting half-space, these are of the form σ(z)=Az^b and μ(z)=Bz^a, where A and B are constants and a+b=-2 or a+b=0. The change in impedance is computed, and the results of numerical simulation are presented     

Fundamental solutions of cavitation in porous solids: a comparative study

The expansion of internally pressurized cavities, embedded in infinite bodies, in spherical and cylindrical (plane strain and plane stress) configurations, is investigated within the framework of finite plasticity. Material response is modeled by the Gurson theory for porous solids and includes strain hardening. Numerical results, obtained under the assumption of nearly universal loading histories, reveal limit cavitation states for all three deformation patterns. Cavitation is identified with asymptotic levels of the specific cavitation energy, which is highest for the spherical cavity and smallest for plane stress (plate) holes. The influence of material porosity is assessed in context of weight optimization of protective plates. A limited comparison with experimental data for porous titanium plate perforation reveals close prediction of ballistic limit velocity.     

Analytical approximations to the solutions for a generalized oscillator with strong nonlinear terms

This paper is focused on solving the generalized second-order strongly nonlinear differential equation ${\ddot{x}+\sum_i {c_i^2 }x \left| x \right|^{i-1}=0}$ which describes the motion of a conservative oscillator with restoring force of series type with integer and noninteger displacement functions. The approximate analytical solution procedures are modified versions of the simple solution approach, the energy balance method, and the frequency?Camplitude formulation including the Petrov?CGalerkin approach. For the case where the linear term is dominant in comparison with the other series terms of the restoring force, the perturbation method based on the solution of the linear differential equation is applied. If the dominant term is nonlinear and the additional terms in the restoring force are small, the perturbation method based on the approximate solution of the pure nonlinear differential equation is introduced. Using the aforementioned methods, the frequency?Camplitude relations in the first approximation are obtained. The suggested solution methods are compared and their advantages and disadvantages discussed. A numerical example is considered, where the restoring force of the oscillator contains a linear and also a noninteger order term (i?=?5/3). The analytically obtained results are compared with numerical results as well as with some approximate analytical results for special cases from the literature.     

Minimum norm solutions to linear elastic analysis problems

A basic problem in the linear elastic analysis is that of finding the vectors of stresses and strains, given a finite element model of a structure and a set of external loads. One purpose of this paper is to show that the problem is a special case of the minimum norm problem for underdetermined systems of linear equations. In this regard, the three conventional structural analysis approaches, i.e. the displacement method, the natural factor formulation and the force method, are unified and interpreted in the framework of the minimum norm problem, which is divided into two approaches—the primal formulation and the dual formulation. Numerical comparisons of several computational procedures capable of solving the minimum norm problem are given from computational efficiency and accuracy points of view. Included in the comparisons are the three conventional structural analysis approaches mentioned above, and several alternative approaches.     

Energy bounds for a mixture of two linear elastic solids occupying a semi-infinite cylinder

Summary The aim of this paper is to establish some forms of the Saint-Venant principle for a mixture of two linear elastic solids occupying a semi-infinite prismatic cylinder. We examine the behaviour of the energy for both static and dynamical problems. Under mild assumptions on the asymptotic behaviour of the unknown fields at infinity, we show that in the static case the elastic energy of the portion of the cylinder beyond a distancex ₃ from the loaded region decays exponentially withx ₃. For the dynamical problem we estimate through the data the total energy stored in that part of the cylinder whose minimum distance from the loaded end isx ₃; these estimates, which are based on the assumption that the initial total energy is finite, depend uponx ₃ but do not depend upon the timet.     

A linear theory for soft ferromagnetic elastic solids

A general theory of magnetoelasticity is developed for soft ferromagnetic materials of multidomain structure, for which the hysteretic loss and exchange effect may be neglected. The general equations are linearized by assuming infinitesimal strains, linear constitutive equations, and that all magnetic variables (magnetic intensity, induction and magnetization) in the deformed body may be divided into two parts: a rigid body state and a perturbation state. The former are the same as those in the magnetostatics for a rigid body and the latter which are the added corrections due to deformations, are coupled with strains and stresses in a set of linear differential equations and boundary conditions. Two versions of the linear theory are given for materials with isotropic, cubic or uniaxial symmetry. One of them is applied to investigate the buckling of an elastic, isotropic plate in a transversally applied uniform magnetic field. The calculated results agree with previous observations.     

Steady interface crack propagation between two viscoelastic standard solids

The problem of a crack propagating steadily in Mode III along the interface of two bonded viscoelastic materials modeled as standard linear solids is investigated. The mathematical method consists of the application of a Fourier transform and the solution of simultaneous Wiener-Hopf equations. The results show that the variation of the stress intensity factor with crack propagation velocity may exhibit different domains of behavior depending upon the relative magnitude of the short and long-time wave speeds of each constituent. The accuracy of the method is checked by a comparison of the general results with special limiting cases.     

Morphology and linear-elastic moduli of random network solids

The effective linear-elastic moduli of disordered network solids are analyzed by voxel-based finite element calculations. We analyze network solids given by Poisson-Voronoi processes and by the structure of collagen fiber networks imaged by confocal microscopy. The solid volume fraction ? is varied by adjusting the fiber radius, while keeping the structural mesh or pore size of the underlying network fixed. For intermediate ?, the bulk and shear modulus are approximated by empirical power-laws K(phi)proptophin and G(phi)proptophim with n≈1.4 and m≈1.7. The exponents for the collagen and the Poisson-Voronoi network solids are similar, and are close to the values n=1.22 and m=2.11 found in a previous voxel-based finite element study of Poisson-Voronoi systems with different boundary conditions. However, the exponents of these empirical power-laws are at odds with the analytic values of n=1 and m=2, valid for low-density cellular structures in the limit of thin beams. We propose a functional form for K(?) that models the cross-over from a power-law at low densities to a porous solid at high densities; a fit of the data to this functional form yields the asymptotic exponent n≈1.00, as expected. Further, both the intensity of the Poisson-Voronoi process and the collagen concentration in the samples, both of which alter the typical pore or mesh size, affect the effective moduli only by the resulting change of the solid volume fraction. These findings suggest that a network solid with the structure of the collagen networks can be modeled in quantitative agreement by a Poisson-Voronoi process.     

Finite cover method for linear and non‐linear analyses of heterogeneous solids

We introduce the finite cover method (FCM) as a generalization of the finite element method (FEM) and extend it to analyse the linear and non‐linear mechanical behaviour of heterogeneous solids and structures. The name ‘FCM’ is actually an alias for the manifold method (MM) and the basic idea of the method has already been established for linear analyses of structures with homogeneous materials. After reviewing the concept of physical and mathematical covers for approximating functions in the FCM, we present the formulation for the static equilibrium state of a structure with arbitrary physical boundaries including material interfaces. The problem essentially involves the discontinuities in strains, and possibly has the discontinuities in displacement caused by interfacial debonding or rupture of material interfaces. We simulate such non‐linear mechanical behaviour after presenting simple numerical examples that demonstrate the equivalence between the approximation capabilities of the FCM and those of the FEM. Copyright © 2003 John Wiley & Sons, Ltd.     

Analytical solutions of sensitivity for pressure microsensors

Pressure microsensors are normally designed in linear operation range. In this study, analytical solutions are presented in order to offer a set of simple equations to designers and researchers to calculate and predict the sensitivity of pressure microsensors. The pressure sensitivity is proportional to the square of the ratio of diaphragm thickness to diaphragm length, but it is inversely proportional to burst pressure. We found that the analytical solutions are in good agreement with simulation by finite element method     

Analytical solutions of plane structures using Mathematica

A study investigating the possibilities of using the program Mathematica to obtain analytical solutions ofplane structures is presented. Two typical applications are developed: plane frames with only bending deformations and plane trusses. In an attempt to keep the number of unknowns as low as possible, the indirect stiffness method and the force method are used to state the first and the second problems, respectively. All the formulations are developed by making use of kernel and generalized inverse matrices, which are also used to characterize the solution of overdetermined systems of equations. Two programs (PlaneFrame and PlaneTruss) were written to solve the first and second problems and the listing of the codes are given in the paper. The results obtained show that problems of a certain complexity can be solved. Copyright © 2001 John Wiley & Sons, Ltd.